How to Explain Integers to Sixth Graders

Until sixth grade, most students work only with positive numbers and zero. Integers open up a whole new world—numbers less than zero! This can be both exciting and confusing. Let's explore how to make negative numbers make sense.

Why Integers Matter for Sixth Graders

Integers aren't just abstract math—they describe real situations:

• Temperature: Below zero readings in winter
• Money: Debt and owing money (negative balance)
• Elevation: Below sea level (Death Valley, ocean depths)
• Football: Yards lost on a play
• Time: Counting down (T-minus 10 seconds)
• Games: Losing points or lives

Understanding integers prepares students for:

• Algebra (solving equations like x + 5 = 3)
• Graphing on coordinate planes
• Understanding scientific measurements
• Financial literacy (debt, profit/loss)

Key Concepts Broken Down Simply

What Are Integers?

Definition: Integers are whole numbers and their opposites, including zero.

Integers: ... -4, -3, -2, -1, 0, 1, 2, 3, 4 ...

├──────────────────┼──────────────────┤
    Negative       Zero    Positive
    Integers               Integers

Not integers: Fractions (1/2), decimals (3.5), or mixed numbers (2¾)

The Number Line

The number line is the most important tool for understanding integers.

        ←  Negative  │  Positive  →
                     │
    ────┼────┼────┼────┼────┼────┼────┼────
       -4   -3   -2   -1    0    1    2    3

    • Numbers get SMALLER as you go LEFT
    • Numbers get LARGER as you go RIGHT
    • Zero is in the middle (neither positive nor negative)

Opposites

Every integer has an opposite—the same distance from zero but in the other direction.

Opposites:
  5 and -5 (both 5 units from zero)
  -12 and 12 (both 12 units from zero)
  0 and 0 (zero is its own opposite!)

On the number line:
    ←── 5 units ──┼── 5 units ──→
                  │
    ──────┼───────┼───────┼──────
         -5       0       5

The opposite of a number can be written as:

• opposite of 5 = -5
• opposite of -3 = 3 (or -(-3) = 3)

Absolute Value

Absolute value is the distance from zero, regardless of direction.

|5| = 5   (5 is 5 units from zero)
|-5| = 5  (-5 is also 5 units from zero)
|0| = 0   (zero is 0 units from zero)

    │← 5 units →│← 5 units →│
    ──────┼──────┼──────┼──────
         -5      0      5
         │             │
         └─ both have absolute value of 5 ─┘

Key insight: Absolute value is always positive (or zero).

Comparing Integers

This is where students often get confused!

Rule: On a number line, numbers to the RIGHT are always GREATER.

Compare: -3 and 2
    ────┼────┼────┼────┼────┼────┼────
       -4   -3   -2   -1    0    1    2
             ↑                        ↑
            -3                        2

2 is to the RIGHT of -3, so 2 > -3 (or -3 < 2)

Common confusion: Students think -10 > -2 because 10 > 2

Reality check:

    ────┼────┼────┼────┼────┼────┼────
      -10  -8   -6   -4   -2    0    2
       ↑                   ↑
     -10                  -2

-2 is to the RIGHT, so -2 > -10
(Think: -2°F is warmer than -10°F)

Ordering Integers

From least to greatest, arrange by position on number line (left to right):

Order: 5, -3, 0, -7, 2

Step 1: Plot on number line
    ────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼────┼────
       -7   -6   -5   -4   -3   -2   -1    0    1    2    3    4    5
        ↑                   ↑                   ↑         ↑         ↑

Step 2: Read left to right
Answer: -7, -3, 0, 2, 5

Visual Examples and Diagrams

Temperature Model

Temperature is the most intuitive integer context for most students.

                  ┌─────┐
                  │ 40° │ Hot summer day
                  ├─────┤
                  │ 30° │
                  ├─────┤
                  │ 20° │ Room temperature
                  ├─────┤
                  │ 10° │
                  ├─────┤
         ──────── │  0° │ ──── Freezing point ────
                  ├─────┤
                  │-10° │
                  ├─────┤
                  │-20° │ Very cold!
                  ├─────┤
                  │-30° │
                  └─────┘

Questions to ask:
• Which is colder: -15° or -5°?  (-15° is colder—further below zero)
• What's the difference between 20° and -10°? (30 degrees)

Elevator/Floor Model

        ┌───────────────┐
    5   │   Penthouse   │
        ├───────────────┤
    4   │   Floor 4     │
        ├───────────────┤
    3   │   Floor 3     │
        ├───────────────┤
    2   │   Floor 2     │
        ├───────────────┤
    1   │   Lobby       │
═══════════════════════════════ Ground Level
   -1   │   Parking P1  │
        ├───────────────┤
   -2   │   Parking P2  │
        ├───────────────┤
   -3   │   Storage     │
        └───────────────┘

• Starting at floor 3, going down 5 floors: 3 + (-5) = -2 (Parking P2)
• Starting at floor -2, going up 4 floors: -2 + 4 = 2 (Floor 2)

Money Model (Profit and Loss)

    Have money (+)     │      Owe money (-)
                       │
    +$50 (savings)     │      -$30 (debt)
    +$25 (earned)      │      -$15 (borrowed)
                       │
         ASSET         │          DEBT
                       │
         ┌─────────────┼─────────────┐
         │             │             │
    ─────┼─────────────┼─────────────┼─────
       -$50          $0            +$50

If you have $20 and spend $35:
$20 + (-$35) = -$15  (You now owe $15)

Football Yards Model

    ← Losing yards (-)   │   Gaining yards (+) →
                         │
    ─────────┼───────────┼───────────┼─────────
           -20          0          +20

    Play 1: Gain 8 yards  →  +8
    Play 2: Lose 3 yards  →  -3
    Play 3: Lose 12 yards →  -12
    Play 4: Gain 5 yards  →  +5

    Total: 8 + (-3) + (-12) + 5 = -2 yards
    (Net loss of 2 yards)

Operations with Integers

Adding Integers

Same signs: Add absolute values, keep the sign

  5 + 3 = 8      (both positive, sum is positive)
 -5 + (-3) = -8  (both negative, sum is negative)

Different signs: Subtract absolute values, keep sign of larger absolute value

  5 + (-3) = 2   (|5| > |-3|, so positive)
 -5 + 3 = -2    (|-5| > |3|, so negative)

Number line visualization:

5 + (-3):
Start at 5, move 3 units LEFT (negative direction)

    ────┼────┼────┼────┼────┼────┼────┼────
        0    1    2    3    4    5    6
                       ↑←───────←↑
                    End at 2    Start at 5

Subtracting Integers

Key insight: Subtracting is adding the opposite!

a - b = a + (-b)

Examples:
  5 - 3 = 5 + (-3) = 2
  5 - (-3) = 5 + 3 = 8
 -5 - 3 = -5 + (-3) = -8
 -5 - (-3) = -5 + 3 = -2

The "keep-change-change" method:

┌─────────────────────────────────────────┐
│         KEEP - CHANGE - CHANGE          │
├─────────────────────────────────────────┤
│   -5  -  (-3)                           │
│                                         │
│   KEEP the first number:     -5         │
│   CHANGE subtraction to addition: +     │
│   CHANGE the sign of second: 3          │
│                                         │
│   -5 + 3 = -2                           │
└─────────────────────────────────────────┘

Multiplying Integers

The sign rules:

┌─────────────┬─────────────┬─────────────┐
│   Signs     │   Example   │   Result    │
├─────────────┼─────────────┼─────────────┤
│   (+)(+)    │   3 × 4     │   +12       │
│   (+)(-)    │   3 × (-4)  │   -12       │
│   (-)(+)    │   (-3) × 4  │   -12       │
│   (-)(-)    │   (-3)×(-4) │   +12       │
└─────────────┴─────────────┴─────────────┘

Summary:
• Same signs → Positive result
• Different signs → Negative result

Pattern explanation for (-)(-)= +:

    3 × (-2) = -6
    2 × (-2) = -4    (+2 from previous)
    1 × (-2) = -2    (+2 from previous)
    0 × (-2) = 0     (+2 from previous)
   -1 × (-2) = 2     (+2 from previous, continuing pattern!)
   -2 × (-2) = 4     (+2 from previous)

Dividing Integers

Same sign rules as multiplication!

Same signs → Positive quotient
  12 ÷ 4 = 3
  (-12) ÷ (-4) = 3

Different signs → Negative quotient
  12 ÷ (-4) = -3
  (-12) ÷ 4 = -3

Hands-On Activities

Activity 1: Integer Card Game

Materials: Deck of cards (red = negative, black = positive; face cards = 10, Ace = 1)

Play:

1. Each player draws two cards
2. Add your integers together
3. Highest sum wins the round
4. Variation: Multiply your cards instead

Activity 2: Temperature Tracking

Materials: Thermometer, weather app, graph paper

Instructions:

1. Record daily high and low temperatures for a week
2. Include negative numbers if in winter
3. Calculate the daily range (high - low)
4. Plot on a number line graph

Activity 3: Elevator Math

Materials: Paper "elevator" with floor numbers (-3 to 10)

Play:

1. Start on a random floor
2. Draw cards showing movements (+3, -5, etc.)
3. Move the elevator
4. Write the equation for each move
5. Example: Start at 4, draw -7: 4 + (-7) = -3

Activity 4: Integer War

Materials: Integer cards from -12 to +12

Play:

1. Deal cards evenly between players
2. Both flip one card
3. Player with the GREATER integer wins both
4. Discuss: Why is 3 greater than -5?

Activity 5: Zero Pairs

Materials: Two-color counters (or coins with different sides)

Concept: Yellow = +1, Red = -1, and (+1) + (-1) = 0

Activity:

1. Model numbers using counters
2. Model addition by combining groups
3. Remove "zero pairs" (one red + one yellow)
4. Count remaining counters

Model: 5 + (-3)

  ○ ○ ○ ○ ○    (5 yellow = +5)
  ● ● ●        (3 red = -3)

Pair up:
  ○ ○ ○ ○ ○
  ● ● ●

  ○-● ○-● ○-● = 0
  ○ ○ remain

Answer: +2

Common Mistakes and How to Fix Them

Mistake 1: Thinking Larger Numbers Are Always Greater

Wrong: -10 > -2 (because 10 > 2)

Correct: -2 > -10

Fix: Always use the number line. "Which is further right?"

Think temperature: Which is warmer, -2° or -10°?
Obviously -2° is warmer (greater)!

Mistake 2: Subtracting Instead of Adding the Opposite

Wrong: 5 - (-3) = 5 - 3 = 2

Correct: 5 - (-3) = 5 + 3 = 8

Fix: Use "keep-change-change" and physically write out the steps.

Mistake 3: Sign Errors in Multiplication

Wrong: (-3) × (-4) = -12

Correct: (-3) × (-4) = +12

Fix: Use the pattern approach to discover why negative × negative = positive.

Mistake 4: Confusing Absolute Value

Wrong: |-7| = -7

Correct: |-7| = 7

Fix: Remember absolute value asks "how far from zero?"—distance is always positive.

Mistake 5: Order of Operations with Negatives

Wrong: -3² = 9

Correct: -3² = -(3²) = -9

But: (-3)² = (-3)×(-3) = 9

Fix: Pay attention to where the negative sign is and what the exponent applies to.

Practice Ideas for Home

Real-World Integer Spotting

• Check weather temperatures (especially in winter)
• Look at sports statistics (yards gained/lost)
• Discuss bank accounts, savings, and spending
• Explore elevation maps (Death Valley, mountains)

Daily Practice Sets

Comparing and Ordering:

Circle the greater integer:
-5 or -2     |    3 or -8     |    -15 or -12

Addition:

-4 + 7 = ?
6 + (-9) = ?
-3 + (-5) = ?

Subtraction:

5 - 8 = ?
-3 - 4 = ?
-2 - (-6) = ?

Multiplication and Division:

(-4) × 5 = ?
(-6) × (-3) = ?
(-20) ÷ 4 = ?
(-24) ÷ (-8) = ?

Story Problems

1. "The temperature was 5°C in the morning and dropped 12 degrees by night. What was the nighttime temperature?"

2. "A submarine at -200 feet rises 75 feet. What is its new depth?"

3. "A football team gained 8 yards, lost 5 yards, lost 3 yards, and gained 12 yards. What was their total change in position?"

Connection to Future Math Concepts

7th Grade: Rational Numbers

Integers extend to all rational numbers:

Integers: -3, 0, 5
Rational: -3.5, 0.75, 5⅓
Same rules apply!

7th-8th Grade: Solving Equations

x + 5 = 3
x = 3 - 5
x = -2

Without integers, this equation would have "no solution"!

8th Grade: Coordinate Plane

    │
  3 │    ● (2,3)
    │
  0 ├────┼────
    │
 -3 │        ● (4,-3)
    │
    └────────────
   -3   0    3

All four quadrants require integer understanding.

High School: Polynomials and Functions

f(x) = x² - 4x + 3
f(-2) = (-2)² - 4(-2) + 3
     = 4 + 8 + 3
     = 15

Quick Reference

┌────────────────────────────────────────────────────┐
│            INTEGER QUICK REFERENCE                 │
├────────────────────────────────────────────────────┤
│ COMPARING: Further RIGHT on number line = GREATER │
│                                                    │
│ ABSOLUTE VALUE: Distance from zero (always ≥ 0)   │
│                                                    │
│ ADDING:                                           │
│   Same signs → Add, keep sign                     │
│   Different → Subtract, keep bigger's sign        │
│                                                    │
│ SUBTRACTING: Add the opposite!                    │
│   a - b = a + (-b)                                │
│                                                    │
│ MULTIPLYING/DIVIDING:                             │
│   Same signs → Positive                           │
│   Different signs → Negative                      │
│                                                    │
│ (-) × (-) = (+)     (+) × (-) = (-)              │
│ (-) ÷ (-) = (+)     (+) ÷ (-) = (-)              │
└────────────────────────────────────────────────────┘

Understanding integers opens the door to algebra and beyond. By connecting negative numbers to real-world contexts like temperature, elevation, and money, students build intuition that makes abstract operations meaningful. With practice and patience, integers become second nature—a powerful tool for describing the full range of mathematical situations.